Решите уравнение 2cos 2x+(cos x-cos 3x)/(cos x+cos 3x)=2.

Задача №17644: Уравнение

Решите уравнение

2 cos 2 x + \frac{cos x - cos 3 x}{cos x + cos 3 x} = 2.

2 cos 2 x + \frac{cos x - cos 3 x}{cos x + cos 3 x} = 2 ⟺ \frac{cos x - cos 3 x}{cos x + cos 3 x} - 4 sin^{2} x = 0 ⟺

⟺ \frac{sin x sin 2 x}{cos x cos 2 x} - 4 sin^{2} x = 0 ⟺ \frac{sin ^{2} x cos x ( 2 - 4 cos 2 x )}{cos x cos 2 x} = 0 ⟺

⟺ [sin x = 0 cos 2 x = \frac{1}{2} ⟺ [x = k π 2 x = \pm \frac{π}{3} + 2 k π, k \in Z ⟺ [x = k π x = \pm \frac{π}{6} + k π, k \in Z .

Решите уравнение

2 cos 2 x + \frac{cos x - cos 3 x}{cos x + cos 3 x} = 2.

2 cos 2 x + \frac{cos x - cos 3 x}{cos x + cos 3 x} = 2 ⟺ \frac{cos x - cos 3 x}{cos x + cos 3 x} - 4 sin^{2} x = 0 ⟺

⟺ \frac{sin x sin 2 x}{cos x cos 2 x} - 4 sin^{2} x = 0 ⟺ \frac{sin ^{2} x cos x ( 2 - 4 cos 2 x )}{cos x cos 2 x} = 0 ⟺

⟺ [sin x = 0 cos 2 x = \frac{1}{2} ⟺ [x = k π 2 x = \pm \frac{π}{3} + 2 k π, k \in Z ⟺ [x = k π x = \pm \frac{π}{6} + k π, k \in Z .